Lesson 1 · 25 min

Light and absorbance: the Beer–Lambert law

Double the concentration and the transmitted light does not halve. Transmittance falls exponentially; absorbance is what is proportional to concentration — and stray light, wide slits and chemical equilibria can bend the line.

In this lesson

Distinguish transmittance T from absorbance A and convert between them
Use the Beer–Lambert law A = εbc to turn an absorbance into a concentration, and know what the molar absorptivity ε means
Explain why it is absorbance, not transmittance, that is proportional to concentration
Recognise the three deviations from Beer’s law: stray light, polychromatic light and chemical equilibria

Hold a glass of potassium permanganate solution up to green light: 1 cm of a 2.0 × 10⁻⁴ mol/L solution lets only about 32 % of 526 nm light through. Double the concentration — does the transmitted light halve to 16 %? Or is it all blocked? Spectrophotometry, one of the most used quantitative methods in any chemistry lab, rests on the answer.

Predict first

A solution transmits 50 % of the light. With the same cell, you double the concentration. What fraction is transmitted now?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

0 % 25 % 50 % 100 %

Transmittance and absorbance

Light of power $P_0$ enters the sample and $P$ comes out. The transmittance is $T = P/P_0$ (often given as %T) and the absorbance is $A = -\lg T = \lg(P_0/P)$ . They map one to one: $T = 100\,\%$ gives $A = 0$ ; $T = 10\,\%$ gives $A = 1$ ; $T = 1\,\%$ gives $A = 2$ . Each extra unit of absorbance leaves a tenth as much light.

Think of the solution as many thin layers, each removing the same fraction of the light that reaches it: the power then decays exponentially along the path, $P = P_0 \cdot 10^{-\varepsilon b c}$ . Taking the logarithm turns the exponent into a product — the Beer–Lambert law.

A = -\lg T = \lg\frac{P_0}{P} = \varepsilon\, b\, c

For several absorbers that do not interact, absorbances add: A = b(ε₁c₁ + ε₂c₂ + …).

$A$: absorbance (dimensionless)
$T$: transmittance P/P₀
$\varepsilon$: molar absorptivity, L mol⁻¹ cm⁻¹; it changes with wavelength and is the substance’s fingerprint
$b$: path length (cell width), cm
$c$: concentration, mol/L

Molar absorptivities of some coloured solutions: allowed transitions are hundreds of times stronger than “forbidden” d–d ones
Species	λmax / nm	ε / (L mol⁻¹ cm⁻¹)	Transition
$\ce{[Cu(H2O)6]^2+}$	800	12	d–d
$\ce{[Cu(NH3)4(H2O)2]^2+}$	605	52	d–d
$\ce{MnO4-}$	526, 546	2.4 × 10³	charge transfer
$\ce{FeSCN^2+}$	447	4.7 × 10³	charge transfer
$\ce{[Fe(phen)3]^2+}$	510	1.11 × 10⁴	charge transfer

Source: Harris, Quantitative Chemical Analysis 9th ed.; Frank & Oswalt (1947); Fortune & Mellon (1938); Housecroft & Sharpe.

Worked example

Mass of permanganate from an absorbance

A KMnO₄ sample is dissolved and made up to 250.0 mL; in a 1.00 cm cell at 526 nm it reads A = 0.612. Taking ε = 2.40 × 10³ L mol⁻¹ cm⁻¹, find the mass of KMnO₄ (M = 158.03 g/mol) and the fraction of light transmitted.

When Beer’s law bends

Beer’s law itself is quite exact for dilute solutions, but measured A–c lines often bend. There are three families of reasons:

Stray light: the detector always receives a little light that skipped the sample or has the wrong wavelength. The darker the sample, the more that light dominates, and the reading tops out near $A_\text{max} = \lg\frac{1+s}{s}$ (s is the stray fraction).
Polychromatic light: the monochromator passes a band of wavelengths (its spectral bandwidth), not a single line. At a band maximum ε hardly changes and little harm is done; on a steep flank ε varies and the line bends down.
Chemical equilibria: if the absorber takes part in an equilibrium that shifts with concentration (dimerisation, acid dissociation, complexation), the share of each form changes with c. An unbuffered indicator bends upward at one wavelength and downward at another, and stays straight only at the isosbestic point, where both forms have the same ε.

Above about 0.01 mol/L, interactions between molecules and changes in refractive index also alter ε itself.

Three deviations, all computed

Why Beer’s law bends

Stray light s = 0.50 % Ceiling A = lg((1+s)/s) = 2.30

Ideal Beer’s law
Instrument reading

At the top concentration: ideal A = 2.940, reading 2.213 (−24.7 %)

A little light always reaches the detector at the wrong wavelength and is not absorbed by the sample. The darker the sample, the more that stray light dominates, and the reading is propped up.

Start with stray light: raise s from 0 to 1 % and the top concentration is held near A ≈ 2. Switch to polychromatic light and compare the 526 nm peak with the 500 nm flank at a 20 nm bandwidth. Finally the chemical case: unbuffered bromocresol green bends down at 617 nm and up at 444 nm, yet is straight at the isosbestic point; add a buffer and all three straighten.

Concept check

You calibrate with an unbuffered acid–base indicator. At which wavelength will A be proportional to the total concentration?

The absorption maximum of the acid form HIn The absorption maximum of the base form In⁻ The isosbestic point, where both forms have the same ε Any wavelength works equally well

Try the calculation

An instrument has 0.50 % stray light (relative to the reference beam). What does it read for a sample whose true absorbance is 2.00?

Key takeaways

T = P/P₀ and A = −lg T; each unit of A cuts the transmitted light tenfold.
Beer–Lambert: A = εbc. Every thin layer absorbs the same fraction of light, so A — not T — is proportional to concentration; absorbances of several absorbers add.
ε is a substance’s fingerprint at a wavelength: 1–100 for d–d transitions, 10³–10⁴ L mol⁻¹ cm⁻¹ for charge transfer.
Stray light depresses high readings; a wide slit on a steep flank bends the line down; chemical equilibria bend it too, except at an isosbestic point.

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Data and references

D. C. Harris, Quantitative Chemical Analysis, 9th ed., W. H. Freeman (2016), ch. 17–18 (Beer’s law, deviations, stray light).
D. A. Skoog, F. J. Holler, S. R. Crouch, Principles of Instrumental Analysis, 7th ed., Cengage (2018), ch. 13 (limitations to Beer’s law, polychromatic radiation, chemical deviations).
Permanganate maxima and ε: G. den Boef et al., Recl. Trav. Chim. Pays-Bas 77 (1958), doi:10.1002/recl.19580771110; Harris, ch. 18.
Indicator bands: the site’s teaching fits for bromocresol green (pKa 4.7), Harris Table 11-4.